Discontinuous nonequilibrium phase transitions in a nonlinearly pulse-coupled excitable lattice model

Vladimir R. V. Assis and Mauro Copelli

Phys. Rev. E 80, 061105 – Published 7 December 2009

Abstract

We study a modified version of the stochastic susceptible-infected-refractory-susceptible (SIRS) model by employing a nonlinear (exponential) reinforcement in the contagion rate and no diffusion. We run simulations for complete and random graphs as well as $d$ -dimensional hypercubic lattices (for $d = 3, 2, 1$ ). For weak nonlinearity, a continuous nonequilibrium phase transition between an absorbing and an active phase is obtained, such as in the usual stochastic SIRS model [Joo and Lebowitz, Phys. Rev. E 70, 036114 (2004)]. However, for strong nonlinearity, the nonequilibrium transition between the two phases can be discontinuous for $d \geq 2$ , which is confirmed by well-characterized hysteresis cycles and bistability. Analytical mean-field results correctly predict the overall structure of the phase diagram. Furthermore, contrary to what was observed in a model of phase-coupled stochastic oscillators with a similar nonlinearity in the coupling [Wood et al., Phys. Rev. Lett. 96, 145701 (2006)], we did not find a transition to a stable (partially) synchronized state in our nonlinearly pulse-coupled excitable elements. For long enough refractory times and high enough nonlinearity, however, the system can exhibit collective excitability and unstable stochastic oscillations.

Received 26 June 2009

DOI:https://doi.org/10.1103/PhysRevE.80.061105

Authors & Affiliations

Vladimir R. V. Assis^* and Mauro Copelli^†

Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil

^*Corresponding author; vladimirassis@df.ufpe.br
^†mcopelli@df.ufpe.br

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 80, Iss. 6 — December 2009

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) Phase space $[P (S), P (I)]$ showing the nullclines $\dot{P} (I) = 0$ (dashed lines) and $\dot{P} (S) = 0$ (dot-dashed lines). Gray triangles corresponds to the physically acceptable region $0 \leq P (I) \leq 1 - P (S)$ . In each row, $a$ is kept constant and $b$ increases from left to right. In the lower panel (b) $(a = 0.5)$ , the absorbing state at $P (S) = 1$ loses stability in a transcritical bifurcation as $b$ increases. In the upper panel (a) $(a = 2.25)$ , the $\dot{P} (I) = 0$ nullcline is bent by the exponential nonlinearity [see Eq. (4)], creating an active state with finite $P (I)$ in a saddle-node bifurcation. Closed (open) symbols denote stable (unstable) fixed points. Note that the nullcline $\dot{P} (I) = 0$ always comprises the line $P (I) = 0$ . When the absorbing state is unstable (a saddle), $P (I) = 0$ corresponds to its stable manifold (rightmost column). Except for this case, the trajectories (solid lines) represent the stable and unstable manifolds of the saddles.Reuse & Permissions
Figure 2
(Color online) Phase diagram of the (mean-field version of the) model. The dashed line marks the onset of phase bistability, with the sudden appearance of a stable active phase with finite order parameter [saddle-node bifurcation in Eqs. (5, 6, 7)]. Solid lines denote a transition in which the absorbing state loses stability [transcritical bifurcation in Eqs. (5, 6, 7)]. Symbols correspond to simulations in a random graph with $γ = 1$ , $K = 10$ , $N = 10^{4}$ , $t_{max} = 5 \times 10^{3}$ , and $h = 10^{- 5}$ averaged over five runs. The lower (upper) inset shows the order parameter $P {(I)}^{*}$ for a cyclic change in the coupling parameter $b$ , showing a continuous (discontinuous) transition for a small (large) value of $a$ . Upward (downward) triangles: increasing (decreasing) $b$ .Reuse & Permissions
Figure 3
(Color online) (a) Phase diagram of the (mean-field version of the) model for $γ = 0.01$ (panels b–g also show single-run simulations of a complete graph with $N = 10^{6}$ ). Bifurcations are as in Fig. 2, except for those additionally occurring within the marked rectangle (inset): for increasing $b$ , a saddle-node bifurcation (dashed line) is followed by a Hopf (H) bifurcation (dotted line), after which a homoclinic (HC) bifurcation occurs (see text). The absorbing phase corresponds to the gray region. The labels (b)–(d) in the inset show the points in parameter space which correspond to the phase portraits below. (b) A saddle and an unstable fixed point are born in a saddle-node bifurcation. (c) The fixed point becomes stable via a Hopf bifurcation and is surrounded by an unstable limit cycle. (d) After a homoclinic bifurcation, the limit cycle disappears. $UUM = upper$ unstable manifold of the saddle; $USM = upper$ stable manifold of the saddle; $ULC = unstable$ limit cycle; $LSM = lower$ stable manifold. The scale in (b) also applies (apart from a translation) to (c) and (d). The lower unstable manifold of the saddle always goes to the absorbing fixed point. Panels e, f, and g show time series for simulated trajectories, respectively, shown in panels b, c, and d, with examples of collective excitability and stochastic oscillations (see text for details).Reuse & Permissions
Figure 4
(Color online) Simulations for $d = 3$ with $N = 25^{3}$ and $h = 2.5 \times 10^{- 5}$ . (a) ( $a = 4.5$ , averaged over 20 runs) The length of the hysteresis cycle $Δ b$ decreases for increasing $t_{max}$ ( $Δ b$ denoted by the horizontal arrow for $t_{max} = 10^{5}$ ). Upward (downward) triangles denote increasing (decreasing) $b$ . Vertical lines show values of $b$ employed in panels (c) and (d). (b) Dependence of $Δ b$ on $t_{max}$ is qualitatively different for $a$ above (filled circles) and below (open circles) $a_{c}$ , respectively, showing a nonzero or zero asymptotic value in the limit $t_{max} \to \infty$ . The lower plot is well fitted by a power law with exponent 0.55. Note that the precision in $Δ b$ is limited by the increment in $b$ employed in the hysteresis cycle ( $δ b = 1.8 \times 10^{- 3}$ in the case $a = 1.875$ ). Panels (c) and (d) show the time evolution of $P (I)$ (averaged over 20 runs) for $a = 4.5$ and different initial conditions [with $N P (I)$ sites in state $I$ at $t = 0$ and the remaining in state $S$ ], showing phase bistability for $b = 3.25$ but not for $b = 7$ .Reuse & Permissions
Figure 5
(Color online) Simulations for $d = 2$ with $N = 150^{2}$ and $h = 2.5 \times 10^{- 5}$ . (a) ( $a = 7$ , averaged over 10 runs) The length of the hysteresis cycle $Δ b$ decreases for increasing $t_{max}$ ( $Δ b$ denoted by the horizontal arrow for $t_{max} = 2 \times 10^{5}$ ). Upward (downward) triangles denote increasing (decreasing) $b$ . Vertical lines show values of $b$ employed in panels (c) and (d). (b) Dependence of $Δ b$ on $t_{max}$ is qualitatively different for $a$ above (filled circles) and below (open circles) $a_{c}$ , respectively, showing a nonzero or zero asymptotic value in the limit $t_{max} \to \infty$ . The lower plot is well fitted by a power law with exponent 0.63. Note that the precision in $Δ b$ is limited by the increment in $b$ employed in the hysteresis cycle ( $δ b = 10^{- 2}$ in the case $a = 4.5$ ). Panels (c) and (d) show the time evolution of $P (I)$ (averaged over 20 runs) for $a = 7$ and different initial conditions [with $N P (I)$ sites in state $I$ at $t = 0$ and the remaining in state $S$ ], showing phase bistability for $b = 4.4$ but not for $b = 21.5$ .Reuse & Permissions
Figure 6
(Color online) Phase diagram of the model in one-, two-, and three-dimensional hypercubic lattices (squares, circles, and triangles, respectively). Open symbols (with dashed lines to guide the eye) mark the onset of phase bistability, with the sudden appearance of a stable active phase with finite order parameter. Filled symbols (with solid lines to guide the eye) denote a transition in which the absorbing state loses stability. The region in between dashed and solid lines correspond to the bistable regime. Lines without symbols show MF results for comparison. Results correspond to simulations with $γ = 1$ and $h = 2.5 \times 10^{- 5}$ . For $d = 1$ , 2, and 3, linear system sizes were $L = 32400$ , $L = 150$ , and $L = 25$ (where $N = L^{d}$ ), the maximum values of $t_{max}$ were $5 \times 10^{5}$ , $2 \times 10^{5}$ , and $10^{5}$ , with results averaged over 10, 10, and 20 runs, respectively.Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics